Готові списки джерел за темами / Pseudo-spectral based global collocation method

Добірка наукової літератури з теми "Pseudo-spectral based global collocation method"

Автор: Grafiati

Опубліковано: 6 вересня 2023

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Статті в журналах з теми "Pseudo-spectral based global collocation method"

Izadi, Mohammad, and Hari Mohan Srivastava. "Applications of Modified Bessel Polynomials to Solve a Nonlinear Chaotic Fractional-Order System in the Financial Market: Domain-Splitting Collocation Techniques." Computation 11, no. 7 (July 3, 2023): 130. http://dx.doi.org/10.3390/computation11070130.

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We propose two accurate and efficient spectral collocation techniques based on a (novel) domain-splitting strategy to handle a nonlinear fractional system consisting of three ODEs arising in financial modeling and with chaotic behavior. One of the major numerical difficulties in designing traditional spectral methods is in the handling of model problems on a long computational domain, which usually yields to loss of accuracy. One remedy is to split the underlying domain and apply the spectral method locally in each subdomain rather than on the global domain of interest. To treat the chaotic financial system numerically, we use the generalized version of modified Bessel polynomials (GMBPs) in the collocation matrix approaches along with the domain-splitting strategy. Whereas the first matrix collocation scheme is directly applied to the financial model problem, the second one is a combination of the quasilinearization method and the direct first numerical matrix method. In the former approach, we arrive at nonlinear algebraic matrix equations while the resulting systems are linear in the latter method and can be solved more efficiently. A convergence theorem related to GMBPs is proved and an upper bound for the error is derived. Several simulation outcomes are provided to show the utility and applicability of the presented matrix collocation procedures.

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Khanduzi, Raheleh, Asyieh Ebrahimzadeh, and Samaneh Panjeh Ali Beik. "Optimal control of fractional integro-differential systems based on a spectral method and grey wolf optimizer." An International Journal of Optimization and Control: Theories & Applications (IJOCTA) 10, no. 1 (January 14, 2020): 55–65. http://dx.doi.org/10.11121/ijocta.01.2020.00753.

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This paper elaborated an effective and robust metaheuristic algorithm with acceptable performance based on solution accuracy. The algorithm applied in solution of the optimal control of fractional Volterra integro-differential (FVID) equation which be substituted by nonlinear programming (NLP). Subsequently the FIVD convert the problem to a NLP by using spectral collocation techniques and thereafter we execute the grey wolf optimizer (GWO) to improve the speed and accuracy and find the solutions of the optimal control and state as well as the optimal value of the cost function. It is mentioned that the utilization of the GWO is simple, due to the fact that the GWO is global search algorithm, the method can be applied to find optimal solution of the NLP. The efficiency of the proposed scheme is shown by the results obtained in comparison with the local methods. Further, some illustrative examples introduced with their approximate solutions and the results of the present approach compared with those achieved using other methods.

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Khader, M. M., and Mohammed M. Babatin. "On Approximate Solutions for Fractional Logistic Differential Equation." Mathematical Problems in Engineering 2013 (2013): 1–7. http://dx.doi.org/10.1155/2013/391901.

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A new approximate formula of the fractional derivatives is derived. The proposed formula is based on the generalized Laguerre polynomials. Global approximations to functions defined on a semi-infinite interval are constructed. The fractional derivatives are presented in terms of Caputo sense. Special attention is given to study the convergence analysis and estimate an error upper bound of the presented formula. The new spectral Laguerre collocation method is presented for solving fractional Logistic differential equation (FLDE). The properties of Laguerre polynomials approximation are used to reduce FLDE to solve a system of algebraic equations which is solved using a suitable numerical method. Numerical results are provided to confirm the theoretical results and the efficiency of the proposed method.

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Ibrahim, Yasser F., Sobhi E. Abd El-Bar, Mohamed M. Khader, and Mohamed Adel. "Studying and Simulating the Fractional COVID-19 Model Using an Efficient Spectral Collocation Approach." Fractal and Fractional 7, no. 4 (April 2, 2023): 307. http://dx.doi.org/10.3390/fractalfract7040307.

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We give a theoretical and numerical analysis of a coronavirus (COVID-19) infection model in this research. A mathematical model of this system is provided, based on a collection of fractional differential equations (in the Caputo sense). Initially, a rough approximation formula was created for the fractional derivative of tp. Here, the third-kind Chebyshev approximations of the spectral collocation method (SCM) were used. To identify the unknown coefficients of the approximate solution, the proposed problem was transformed into a system of algebraic equations, which was then transformed into a restricted optimization problem. To evaluate the effectiveness and accuracy of the suggested scheme, the residual error function was computed. The objective of this research was to halt the global spread of a disease. A susceptible person may be moved immediately into the confined class after being initially quarantined or an exposed person may be transferred to one of the infected classes. The researchers adopted this strategy and considered both asymptomatic and symptomatic infected patients. Results acquired with the achieved results were contrasted with those obtained using the generalized Runge-Kutta method.

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CHAKRABORTY, DEBANANDA, JAE-HUN JUNG, and GAURAV KHANNA. "A MULTI-DOMAIN HYBRID METHOD FOR HEAD-ON COLLISION OF BLACK HOLES IN PARTICLE LIMIT." International Journal of Modern Physics C 22, no. 05 (May 2011): 517–41. http://dx.doi.org/10.1142/s0129183111016415.

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A hybrid method is developed based on the spectral and finite-difference methods for solving the inhomogeneous Zerilli equation in time-domain. The developed hybrid method decomposes the domain into the spectral and finite-difference domains. The singular source term is located in the spectral domain while the solution in the region without the singular term is approximated by the higher-order finite-difference method. The spectral domain is also split into multi-domains and the finite-difference domain is placed as the boundary domain. Due to the global nature of the spectral method, a multi-domain method composed of the spectral domain only does not yield the proper power-law decay unless the range of the computational domain is large. The finite-difference domain helps reduce boundary effects due to the truncation of the computational domain. The multi-domain approach with the finite-difference boundary domain method reduces the computational cost significantly and also yields the proper power-law decay. Stable and accurate interface conditions between the finite-difference and spectral domains and the spectral and spectral domains are derived. For the singular source term, we use both the Gaussian model with various values of full width at half-maximum and a localized discrete δ-function. The discrete δ-function was generalized to adopt the Gauss–Lobatto collocation points of the spectral domain. The gravitational waveforms are measured. Numerical results show that the developed hybrid method accurately yields the quasi-normal modes and the power-law decay profile. The numerical results also show that the power-law decay profile is less sensitive to the shape of the regularized δ-function for the Gaussian model than expected. The Gaussian model also yields better results than the localized discrete δ-function.

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Sánchez, José L. Carrión, Sílvio R. C. de Freitas, and Riccardo Barzaghi. "OFFSET EVALUATION OF THE ECUADORIAN VERTICAL DATUM RELATED TO THE IHRS." Boletim de Ciências Geodésicas 24, no. 4 (December 2018): 503–25. http://dx.doi.org/10.1590/s1982-21702018000400031.

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Abstract Considering the definition of the International Height Reference System (IHRS) in the geopotential space (Resolution 1/2015, International Association of Geodesy - IAG), among the present main objectives of the international geodetic community is the materialization of IHRS around the world. One fundamental task for this is the offset determination of each national vertical datum related to the IHRS. In this manuscript we establish the relationship between the Ecuadorian Vertical Datum (EVD) and the IHRS in the geopotential space following the foundations of the Resolution 1/2015 IAG. Gravity data, heights from the Ecuadorian Fundamental Vertical Network, Global Geopotential Models and Digital Elevation Models were used in the computations. Based on the Least Squares Collocation method, empirical covariance functions and spectral decomposition techniques, we realized the modelling of the geopotential in the study region (4° x 4° centered in the La Libertad tide gauge, Ecuador). Based on the referred approaches, we solved the free Geodetic Boundary Value Problem for determining the discrepancy of the EVD related to the IHRS. An offset of approximately 29 cm ± 3 cm was estimated for the W 0 - W 0 i relation when the GO_CONS_GCF_2_DIR_R5 model was used in the modeling of the medium and long wavelengths of the terrestrial gravity field, and approximately 43 cm ± 3 cm when the EIGEN6C4 model was used.

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Kim, Chang Ho, and U. Jin Choi. "Spectral collocation methods for a partial integro-differential equation with a weakly singular kernel." Journal of the Australian Mathematical Society. Series B. Applied Mathematics 39, no. 3 (January 1998): 408–30. http://dx.doi.org/10.1017/s0334270000009474.

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AbstractWe propose and analyze the spectral collocation approximation for the partial integro-differential equations with a weakly singular kernel. The space discretization is based on the pseudo-spectral method, which is a collocation method at the Gauss-Lobatto quadrature points. We prove unconditional stability and obtain the optimal error bounds which depend on the time step, the degree of polynomial and the Sobolev regularity of the solution.

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Lovetskiy, Konstantin P., Dmitry S. Kulyabov, and Ali Weddeye Hissein. "Multistage pseudo-spectral method (method of collocations) for the approximate solution of an ordinary differential equation of the first order." Discrete and Continuous Models and Applied Computational Science 30, no. 2 (May 3, 2022): 127–38. http://dx.doi.org/10.22363/2658-4670-2022-30-2-127-138.

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The classical pseudospectral collocation method based on the expansion of the solution in a basis of Chebyshev polynomials is considered. A new approach to constructing systems of linear algebraic equations for solving ordinary differential equations with variable coefficients and with initial (and/or boundary) conditions makes possible a significant simplification of the structure of matrices, reducing it to a diagonal form. The solution of the system is reduced to multiplying the matrix of values of the Chebyshev polynomials on the selected collocation grid by the vector of values of the function describing the given derivative at the collocation points. The subsequent multiplication of the obtained vector by the two-diagonal spectral matrix, inverse with respect to the Chebyshev differentiation matrix, yields all the expansion coefficients of the sought solution except for the first one. This first coefficient is determined at the second stage based on a given initial (and/or boundary) condition. The novelty of the approach is to first select a class (set) of functions that satisfy the differential equation, using a stable and computationally simple method of interpolation (collocation) of the derivative of the future solution. Then the coefficients (except for the first one) of the expansion of the future solution are determined in terms of the calculated expansion coefficients of the derivative using the integration matrix. Finally, from this set of solutions only those that correspond to the given initial conditions are selected.

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SEO, JEONG-KWEON, and BYEONG-CHUN SHIN. "NUMERICAL SOLUTIONS OF BURGERS EQUATION BY REDUCED-ORDER MODELING BASED ON PSEUDO-SPECTRAL COLLOCATION METHOD." Journal of the Korea Society for Industrial and Applied Mathematics 19, no. 2 (June 25, 2015): 123–35. http://dx.doi.org/10.12941/jksiam.2015.19.123.

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Bhrawy, A. H., M. A. Alghamdi, and D. Baleanu. "Numerical Solution of a Class of Functional-Differential Equations Using Jacobi Pseudospectral Method." Abstract and Applied Analysis 2013 (2013): 1–9. http://dx.doi.org/10.1155/2013/513808.

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The shifted Jacobi-Gauss-Lobatto pseudospectral (SJGLP) method is applied to neutral functional-differential equations (NFDEs) with proportional delays. The proposed approximation is based on shifted Jacobi collocation approximation with the nodes of Gauss-Lobatto quadrature. The shifted Legendre-Gauss-Lobatto Pseudo-spectral and Chebyshev-Gauss-Lobatto Pseudo-spectral methods can be obtained as special cases of the underlying method. Moreover, the SJGLP method is extended to numerically approximate the nonlinear high-order NFDE with proportional delay. Some examples are displayed for implicit and explicit forms of NFDEs to demonstrate the computation accuracy of the proposed method. We also compare the performance of the method with variational iteration method, one-legθ-method, continuous Runge-Kutta method, and reproducing kernel Hilbert space method.

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Більше джерел

Дисертації з теми "Pseudo-spectral based global collocation method"

Swaminathan, Shriram. "Real Time Solutions to Aerospace Optimal Control Problems." Thesis, 2020. https://etd.iisc.ac.in/handle/2005/5082.

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A real-time autonomous trajectory generation methodology for aerospace vehicles using pseudo-spectral based global collocation methods is presented in this thesis. The proposed method is formulated in a manner whereby it can converge from random initial guess pro les in real time. The proposed methodology is validated using Monte-Carlo simulations for various practical aerospace problems to demonstrate the robustness in presence of initial condition errors. In the thesis, the solution to the trajectory optimization using both Direct and In- direct methods are presented. A two level optimization scheme is adopted for solving the trajectory and minimizing the fuel consumption using the Direct method . In the outer level, the required burn duration for achieving mission objectives is iterated using a simple optimization procedure for minimizing fuel consumption. In the inner level, the trajectory for a given burn time obtained from the outer level, is solved using a modi ed Newton-Raphson scheme. An important advantage of this scheme is that, it does not use any general purpose, commercially available gradient-based optimization routines as they are not suitable for real-time implementation. In the second part of the thesis, the proposed algorithm is applied to general optimal control problems involving states and co-states. In this scheme, the optimal solution for the states, control and co-states are computed using a discretized version of the Indirect method. The proposed method can handle inequality constraints on states and control as well. The effectiveness of the algorithm is demonstrated for a set of general optimal control problems with emphasis on aerospace trajectory optimization

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Тези доповідей конференцій з теми "Pseudo-spectral based global collocation method"

Zhang, Ce, Wei Ma, Wensheng Yu, and Jinfang Teng. "Effect of Heat Transfer on Pipe Flow Stability." In ASME Turbo Expo 2017: Turbomachinery Technical Conference and Exposition. American Society of Mechanical Engineers, 2017. http://dx.doi.org/10.1115/gt2017-64451.

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The compressibility of flow field has an important effect on flow stability. However, when the compressibility is considered, the effect of Mach number is often considered while the effect of heat transfer is always neglected in the existing flow stability studies. Linear stability analysis tools based on compressible Orr-Sommerfeld (O-S) equations and linearized Navier-Stokes equations in cylindrical coordinate system are established in this paper. These equations are numerically solved by using Chebyshev spectral collocation method and pseudo-modes are eliminated. Linear stability analysis of pipe flow with heat transfer whose average flow field is obtained by CFD simulation is carried out. The results show that for spatial modes, the heating effect of the wall makes pipe flow more unstable, while cooling effect of the wall makes pipe flow more stable. For global modes of pipe flow, the frequency of global mode decreases when the wall cools the flow and the decrease of mean temperature of pipe flow leads to the improvement of global mode stability.

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Chen, He, Sun Dakun, and Sun Xiaofeng. "Stall Inception Analysis of Transonic Compressors With Chordwise and Axial Sweep." In ASME Turbo Expo 2017: Turbomachinery Technical Conference and Exposition. American Society of Mechanical Engineers, 2017. http://dx.doi.org/10.1115/gt2017-63174.

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This paper concentrates on the stall inception analysis of transonic compressors with chordwise and axial sweep. A new prediction approach of stall inception is developed based on global stability analysis and immersed boundary theory, which makes it possible to takes both the concrete blade geometry and complicated base flow into consideration. The prediction of stall inception boils down to an eigenvalue problem. Spectral collocation method is adopted to discretize the eigenvalue equations and the eigenvalues are solved by using singular value decomposition method. The developed prediction approach is validated on two different typical transonic compressors, a single stage compressor and an isolated-rotor compressor, which shows a good agreement with the experimental data. The latter is adopted as a baseline rotor for the investigation of chordwise and axial sweep. By adjusting the stacking line of the baseline rotor, a series of swept rotors are modelled and the stall inception behavior of them are predicted by using the developed approach. The comparison of stall inception behaviors between these rotors is presented, and in combination with steady flow analyses, the effects of sweep features on the stall inception in transonic compressors are discussed.

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Kaloni, P. N., and J. X. Lou. "Convective Stability of Hadley Circulations in Finite Domains." In ASME 2010 3rd Joint US-European Fluids Engineering Summer Meeting collocated with 8th International Conference on Nanochannels, Microchannels, and Minichannels. ASMEDC, 2010. http://dx.doi.org/10.1115/fedsm-icnmm2010-31228.

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We present numerical solutions of natural convection in a cavity of finite domain when the flow is driven by a horizontal temperature gradient between isothermal vertical walls. Only the transverse roll structure of convection is calculated by employing the pseudo spectral Chebyshev collocation method. A numerical solution of the steady flow is first obtained and, based upon it, the critical Rayleigh number (related to Grashof number) of linear transverse mode is determined. The influence of aspect ratio, on the flow and temperature patterns, is also investigated.

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Iklódi, Zsolt, Xavier Beudaert, and Zoltan Dombovari. "On the Modelling Bases of In-Motion Dynamic Characterization of Flexible Structures Subject to Friction and Position Control Delay." In ASME 2022 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. American Society of Mechanical Engineers, 2022. http://dx.doi.org/10.1115/detc2022-90924.

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Abstract This article presents a characterisation technique of in-motion machine dynamics based on the principles of numerical continuation. A linear two degree of freedom mechanical model is considered, representing e.g. a flexible moving column of a machine tool, and is subjected to a non-smooth friction and a delayed feedback drive control force, resulting in a model governed by a system of piecewise-smooth delay differential equations. By applying harmonic forcing to the system, periodic solutions can be found, through the continuation of which, an accurate vibratory characterisation of in-motion machine dynamics can be acquired. In the continuation routine, spectral collocation techniques were employed to formulate the discretized boundary value problem of piecewise-smooth periodic orbits, and the pseudo-arclength method was implemented. Special care was attributed to the detection of grazing and sliding bifurcations, and the continuation routine was also extended to allow continuation of these critical points.

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Babaee, Hessam, Sumanta Acharya, and Xiaoliang Wan. "Optimization of Forcing Parameters of Film Cooling Effectiveness." In ASME Turbo Expo 2013: Turbine Technical Conference and Exposition. American Society of Mechanical Engineers, 2013. http://dx.doi.org/10.1115/gt2013-95636.

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An optimization strategy is described that combines high-fidelity simulations with response surface construction, and is applied to pulsed film cooling for turbine blades. The response surface is constructed for the film cooling effectiveness as a function of duty cycle, in the range of DC between 0.05 and 1, and pulsation frequency St in the range of 0.2–2, using a pseudo-spectral projection method. The jet is fully modulated and the blowing ratio, when the jet is on, is 1.5 in all cases. Overall 73 direct numerical simulations (DNS) using spectral element method were performed to sample the film cooling effectiveness on a Clenshaw-Curtis grid in the design space. The geometry includes a 35-degree delivery tube and a plenum. It is observed that in the parameter space explored a global optimum exists, and in the present study, the best film cooling effectiveness is found at DC = 0.14 and St = 1.03. In the same range of DC and St, four other local optimums were found. The physical mechanisms leading to the forcing parameters of the global optimum are explored and ingestion of the crossflow into the delivery tube is observed to play an important role in this process. The gradient-based optimization algorithms are argued to be unsuitable for the current problem due to the non-convexity of the objective function.

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